The graphs of x^2+y^2=4+12x+6y and x^2+y^2=k+4x+12y intersect when k satisfies a≤ k≤ b, and for no other values of k. Find b-a.

1998 AMC 12 Problem 23

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23.

The graphs of

x^2+y^2=4+12x+6y

and

x^2+y^2=k+4x+12y

intersect when

k

satisfies

a\le k\le b,

and for no other values of

k.

Find

b-a.

5

68

104

140

144

Complete the square to find both circle centers and radii

Two circles intersect exactly when the center distance lies between the difference and sum of their radii

The circles are

(x-6)^2+(y-3)^2=49,

(x-2)^2+(y-6)^2=k+40.

Their centers are

5

units apart. With second radius

r=\sqrt{k+40},

intersection requires

|7-r|\le5\le7+r,

2\le r\le12.

Thus

-36\le k\le104,

and

b-a=140.

The correct answer is D.